lib/ext/qcbor/src/ieee754.c - TF-M/trusted-firmware-m.git - TrustedFirmware Git Browser

 /*==============================================================================
  ieee754.c -- floating point conversion between half, double and single precision

  Copyright (c) 2018-2019, Laurence Lundblade. All rights reserved.

  SPDX-License-Identifier: BSD-3-Clause

  See BSD-3-Clause license in README.md

  Created on 7/23/18
  ==============================================================================*/

 #include "ieee754.h"
 #include <string.h> // For memcpy()


 /*
  This code is written for clarity and verifiability, not for size, on the assumption
  that the optimizer will do a good job. The LLVM optimizer, -Os, does seem to do the
  job and the resulting object code is smaller from combining code for the many different
  cases (normal, subnormal, infinity, zero...) for the conversions.

  Dead stripping is also really helpful to get code size down when floating point
  encoding is not needed.

  This code works solely using shifts and masks and thus has no dependency on
  any math libraries. It can even work if the CPU doesn't have any floating
  point support, though that isn't the most useful thing to do.

  The memcpy() dependency is only for CopyFloatToUint32() and friends which only
  is needed to avoid type punning when converting the actual float bits to
  an unsigned value so the bit shifts and masks can work.
  */

 /*
  The references used to write this code:

  - IEEE 754-2008, particularly section 3.6 and 6.2.1

  - https://en.wikipedia.org/wiki/IEEE_754 and subordinate pages

  - https://stackoverflow.com/questions/19800415/why-does-ieee-754-reserve-so-many-nan-values
  */


 // ----- Half Precsion -----------
 #define HALF_NUM_SIGNIFICAND_BITS (10)
 #define HALF_NUM_EXPONENT_BITS    (5)
 #define HALF_NUM_SIGN_BITS        (1)

 #define HALF_SIGNIFICAND_SHIFT    (0)
 #define HALF_EXPONENT_SHIFT       (HALF_NUM_SIGNIFICAND_BITS)
 #define HALF_SIGN_SHIFT           (HALF_NUM_SIGNIFICAND_BITS + HALF_NUM_EXPONENT_BITS)

 #define HALF_SIGNIFICAND_MASK     (0x3ff) // The lower 10 bits  // 0x03ff
 #define HALF_EXPONENT_MASK        (0x1f << HALF_EXPONENT_SHIFT) // 0x7c00 5 bits of exponent
 #define HALF_SIGN_MASK            (0x01 << HALF_SIGN_SHIFT) //  // 0x80001 bit of sign
 #define HALF_QUIET_NAN_BIT        (0x01 << (HALF_NUM_SIGNIFICAND_BITS-1)) // 0x0200

 /* Biased    Biased    Unbiased   Use
     0x00       0        -15       0 and subnormal
     0x01       1        -14       Smallest normal exponent
     0x1e      30         15       Largest normal exponent
     0x1F      31         16       NaN and Infinity  */
 #define HALF_EXPONENT_BIAS        (15)
 #define HALF_EXPONENT_MAX         (HALF_EXPONENT_BIAS)    //  15 Unbiased
 #define HALF_EXPONENT_MIN         (-HALF_EXPONENT_BIAS+1) // -14 Unbiased
 #define HALF_EXPONENT_ZERO        (-HALF_EXPONENT_BIAS)   // -15 Unbiased
 #define HALF_EXPONENT_INF_OR_NAN  (HALF_EXPONENT_BIAS+1)  //  16 Unbiased


 // ------ Single Precision --------
 #define SINGLE_NUM_SIGNIFICAND_BITS (23)
 #define SINGLE_NUM_EXPONENT_BITS    (8)
 #define SINGLE_NUM_SIGN_BITS        (1)

 #define SINGLE_SIGNIFICAND_SHIFT    (0)
 #define SINGLE_EXPONENT_SHIFT       (SINGLE_NUM_SIGNIFICAND_BITS)
 #define SINGLE_SIGN_SHIFT           (SINGLE_NUM_SIGNIFICAND_BITS + SINGLE_NUM_EXPONENT_BITS)

 #define SINGLE_SIGNIFICAND_MASK     (0x7fffffUL) // The lower 23 bits
 #define SINGLE_EXPONENT_MASK        (0xffUL << SINGLE_EXPONENT_SHIFT) // 8 bits of exponent
 #define SINGLE_SIGN_MASK            (0x01UL << SINGLE_SIGN_SHIFT) // 1 bit of sign
 #define SINGLE_QUIET_NAN_BIT        (0x01UL << (SINGLE_NUM_SIGNIFICAND_BITS-1))

 /* Biased  Biased   Unbiased  Use
     0x0000     0     -127      0 and subnormal
     0x0001     1     -126      Smallest normal exponent
     0x7f     127        0      1
     0xfe     254      127      Largest normal exponent
     0xff     255      128      NaN and Infinity  */
 #define SINGLE_EXPONENT_BIAS        (127)
 #define SINGLE_EXPONENT_MAX         (SINGLE_EXPONENT_BIAS)    // 127  unbiased
 #define SINGLE_EXPONENT_MIN         (-SINGLE_EXPONENT_BIAS+1) // -126 unbiased
 #define SINGLE_EXPONENT_ZERO        (-SINGLE_EXPONENT_BIAS)   // -127 unbiased
 #define SINGLE_EXPONENT_INF_OR_NAN  (SINGLE_EXPONENT_BIAS+1)  // 128  unbiased


 // --------- Double Precision ----------
 #define DOUBLE_NUM_SIGNIFICAND_BITS (52)
 #define DOUBLE_NUM_EXPONENT_BITS    (11)
 #define DOUBLE_NUM_SIGN_BITS        (1)

 #define DOUBLE_SIGNIFICAND_SHIFT    (0)
 #define DOUBLE_EXPONENT_SHIFT       (DOUBLE_NUM_SIGNIFICAND_BITS)
 #define DOUBLE_SIGN_SHIFT           (DOUBLE_NUM_SIGNIFICAND_BITS + DOUBLE_NUM_EXPONENT_BITS)

 #define DOUBLE_SIGNIFICAND_MASK     (0xfffffffffffffULL) // The lower 52 bits
 #define DOUBLE_EXPONENT_MASK        (0x7ffULL << DOUBLE_EXPONENT_SHIFT) // 11 bits of exponent
 #define DOUBLE_SIGN_MASK            (0x01ULL << DOUBLE_SIGN_SHIFT) // 1 bit of sign
 #define DOUBLE_QUIET_NAN_BIT        (0x01ULL << (DOUBLE_NUM_SIGNIFICAND_BITS-1))


 /* Biased      Biased   Unbiased  Use
    0x00000000     0     -1023     0 and subnormal
    0x00000001     1     -1022     Smallest normal exponent
    0x000007fe  2046      1023     Largest normal exponent
    0x000007ff  2047      1024     NaN and Infinity  */
 #define DOUBLE_EXPONENT_BIAS        (1023)
 #define DOUBLE_EXPONENT_MAX         (DOUBLE_EXPONENT_BIAS)    // unbiased
 #define DOUBLE_EXPONENT_MIN         (-DOUBLE_EXPONENT_BIAS+1) // unbiased
 #define DOUBLE_EXPONENT_ZERO        (-DOUBLE_EXPONENT_BIAS)   // unbiased
 #define DOUBLE_EXPONENT_INF_OR_NAN  (DOUBLE_EXPONENT_BIAS+1)  // unbiased


 /*
  Convenient functions to avoid type punning, compiler warnings and such
  The optimizer reduces them to a simple assignment.
  This is a crusty corner of C. It shouldn't be this hard.

  These are also in UsefulBuf.h under a different name. They are copied
  here to avoid a dependency on UsefulBuf.h. There is no
  object code size impact because these always optimze down to a
  simple assignment.
  */
 static inline uint32_t CopyFloatToUint32(float f)
 {
     uint32_t u32;
     memcpy(&u32, &f, sizeof(uint32_t));
     return u32;
 }

 static inline uint64_t CopyDoubleToUint64(double d)
 {
     uint64_t u64;
     memcpy(&u64, &d, sizeof(uint64_t));
     return u64;
 }

 static inline float CopyUint32ToFloat(uint32_t u32)
 {
     float f;
     memcpy(&f, &u32, sizeof(uint32_t));
     return f;
 }

 static inline double CopyUint64ToDouble(uint64_t u64)
 {
     double d;
     memcpy(&d, &u64, sizeof(uint64_t));
     return d;
 }


 // Public function; see ieee754.h
 uint16_t IEEE754_FloatToHalf(float f)
 {
     // Pull the three parts out of the single-precision float
     const uint32_t uSingle = CopyFloatToUint32(f);
     const int32_t  nSingleUnbiasedExponent = ((uSingle & SINGLE_EXPONENT_MASK) >> SINGLE_EXPONENT_SHIFT) - SINGLE_EXPONENT_BIAS;
     const uint32_t uSingleSign             =  (uSingle & SINGLE_SIGN_MASK) >> SINGLE_SIGN_SHIFT;
     const uint32_t uSingleSignificand      =   uSingle & SINGLE_SIGNIFICAND_MASK;


     // Now convert the three parts to half-precision.
     uint16_t uHalfSign, uHalfSignificand, uHalfBiasedExponent;
     if(nSingleUnbiasedExponent == SINGLE_EXPONENT_INF_OR_NAN) {
         // +/- Infinity and NaNs -- single biased exponent is 0xff
         uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
         if(!uSingleSignificand) {
             // Infinity
             uHalfSignificand = 0;
         } else {
             // Copy the LBSs of the NaN payload that will fit from the single to the half
             uHalfSignificand = uSingleSignificand & (HALF_SIGNIFICAND_MASK & ~HALF_QUIET_NAN_BIT);
             if(uSingleSignificand & SINGLE_QUIET_NAN_BIT) {
                 // It's a qNaN; copy the qNaN bit
                 uHalfSignificand |= HALF_QUIET_NAN_BIT;
             } else {
                 // It's a sNaN; make sure the significand is not zero so it stays a NaN
                 // This is needed because not all significand bits are copied from single
                 if(!uHalfSignificand) {
                     // Set the LSB. This is what wikipedia shows for sNAN.
                     uHalfSignificand |= 0x01;
                 }
             }
         }
     } else if(nSingleUnbiasedExponent == SINGLE_EXPONENT_ZERO) {
         // 0 or a subnormal number  -- singled biased exponent is 0
         uHalfBiasedExponent = 0;
         uHalfSignificand    = 0; // Any subnormal single will be too small to express as a half precision
     } else if(nSingleUnbiasedExponent > HALF_EXPONENT_MAX) {
         // Exponent is too large to express in half-precision; round up to infinity
         uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
         uHalfSignificand    = 0;
     } else if(nSingleUnbiasedExponent < HALF_EXPONENT_MIN) {
         // Exponent is too small to express in half-precision normal; make it a half-precision subnormal
         uHalfBiasedExponent = (uint16_t)(HALF_EXPONENT_ZERO + HALF_EXPONENT_BIAS);
         // Difference between single normal exponent and the base exponent of a half subnormal
         const uint32_t nExpDiff = -(nSingleUnbiasedExponent - HALF_EXPONENT_MIN);
         // Also have to shift the significand by the difference in number of bits between a single and a half significand
         const int32_t nSignificandBitsDiff = SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS;
         // Add in the 1 that is implied in the significand of a normal number; it needs to be present in a subnormal
         const uint32_t uSingleSignificandSubnormal = uSingleSignificand + (0x01L << SINGLE_NUM_SIGNIFICAND_BITS);
         uHalfSignificand = uSingleSignificandSubnormal >> (nExpDiff + nSignificandBitsDiff);
     } else {
         // The normal case
         uHalfBiasedExponent = nSingleUnbiasedExponent + HALF_EXPONENT_BIAS;
         uHalfSignificand    = uSingleSignificand >> (SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
     }
     uHalfSign = uSingleSign;

     // Put the 3 values in the right place for a half precision
     const uint16_t uHalfPrecision =  uHalfSignificand |
                                     (uHalfBiasedExponent << HALF_EXPONENT_SHIFT) |
                                     (uHalfSign << HALF_SIGN_SHIFT);
     return uHalfPrecision;
 }


 // Public function; see ieee754.h
 uint16_t IEEE754_DoubleToHalf(double d)
 {
     // Pull the three parts out of the double-precision float
     const uint64_t uDouble = CopyDoubleToUint64(d);
     const int64_t  nDoubleUnbiasedExponent = ((uDouble & DOUBLE_EXPONENT_MASK) >> DOUBLE_EXPONENT_SHIFT) - DOUBLE_EXPONENT_BIAS;
     const uint64_t uDoubleSign             =  (uDouble & DOUBLE_SIGN_MASK) >> DOUBLE_SIGN_SHIFT;
     const uint64_t uDoubleSignificand      =   uDouble & DOUBLE_SIGNIFICAND_MASK;


     // Now convert the three parts to half-precision.
     uint16_t uHalfSign, uHalfSignificand, uHalfBiasedExponent;
     if(nDoubleUnbiasedExponent == DOUBLE_EXPONENT_INF_OR_NAN) {
         // +/- Infinity and NaNs -- single biased exponent is 0xff
         uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
         if(!uDoubleSignificand) {
             // Infinity
             uHalfSignificand = 0;
         } else {
             // Copy the LBSs of the NaN payload that will fit from the double to the half
             uHalfSignificand = uDoubleSignificand & (HALF_SIGNIFICAND_MASK & ~HALF_QUIET_NAN_BIT);
             if(uDoubleSignificand & DOUBLE_QUIET_NAN_BIT) {
                 // It's a qNaN; copy the qNaN bit
                 uHalfSignificand |= HALF_QUIET_NAN_BIT;
             } else {
                 // It's an sNaN; make sure the significand is not zero so it stays a NaN
                 // This is needed because not all significand bits are copied from single
                 if(!uHalfSignificand) {
                     // Set the LSB. This is what wikipedia shows for sNAN.
                     uHalfSignificand |= 0x01;
                 }
             }
         }
     } else if(nDoubleUnbiasedExponent == DOUBLE_EXPONENT_ZERO) {
         // 0 or a subnormal number  -- double biased exponent is 0
         uHalfBiasedExponent = 0;
         uHalfSignificand    = 0; // Any subnormal single will be too small to express as a half precision; TODO, is this really true?
     } else if(nDoubleUnbiasedExponent > HALF_EXPONENT_MAX) {
         // Exponent is too large to express in half-precision; round up to infinity; TODO, is this really true?
         uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
         uHalfSignificand    = 0;
     } else if(nDoubleUnbiasedExponent < HALF_EXPONENT_MIN) {
         // Exponent is too small to express in half-precision; round down to zero
         uHalfBiasedExponent = (uint16_t)(HALF_EXPONENT_ZERO + HALF_EXPONENT_BIAS);
         // Difference between double normal exponent and the base exponent of a half subnormal
         const uint64_t nExpDiff = -(nDoubleUnbiasedExponent - HALF_EXPONENT_MIN);
         // Also have to shift the significand by the difference in number of bits between a double and a half significand
         const int64_t nSignificandBitsDiff = DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS;
         // Add in the 1 that is implied in the significand of a normal number; it needs to be present in a subnormal
         const uint64_t uDoubleSignificandSubnormal = uDoubleSignificand + (0x01ULL << DOUBLE_NUM_SIGNIFICAND_BITS);
         uHalfSignificand = uDoubleSignificandSubnormal >> (nExpDiff + nSignificandBitsDiff);
     } else {
         // The normal case
         uHalfBiasedExponent = nDoubleUnbiasedExponent + HALF_EXPONENT_BIAS;
         uHalfSignificand    = uDoubleSignificand >> (DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
     }
     uHalfSign = uDoubleSign;


     // Put the 3 values in the right place for a half precision
     const uint16_t uHalfPrecision =  uHalfSignificand |
                                     (uHalfBiasedExponent << HALF_EXPONENT_SHIFT) |
                                     (uHalfSign << HALF_SIGN_SHIFT);
     return uHalfPrecision;
 }


 // Public function; see ieee754.h
 float IEEE754_HalfToFloat(uint16_t uHalfPrecision)
 {
     // Pull out the three parts of the half-precision float
     const uint16_t uHalfSignificand      =   uHalfPrecision & HALF_SIGNIFICAND_MASK;
     const int16_t  nHalfUnBiasedExponent = ((uHalfPrecision & HALF_EXPONENT_MASK) >> HALF_EXPONENT_SHIFT) - HALF_EXPONENT_BIAS;
     const uint16_t uHalfSign             =  (uHalfPrecision & HALF_SIGN_MASK) >> HALF_SIGN_SHIFT;


     // Make the three parts of the single-precision number
     uint32_t uSingleSignificand, uSingleSign, uSingleBiasedExponent;
     if(nHalfUnBiasedExponent == HALF_EXPONENT_ZERO) {
         // 0 or subnormal
         if(uHalfSignificand) {
             // Subnormal case
             uSingleBiasedExponent = -HALF_EXPONENT_BIAS + SINGLE_EXPONENT_BIAS +1;
             // A half-precision subnormal can always be converted to a normal single-precision float because the ranges line up
             uSingleSignificand = uHalfSignificand;
             // Shift bits from right of the decimal to left, reducing the exponent by 1 each time
             do {
                 uSingleSignificand <<= 1;
                 uSingleBiasedExponent--;
             } while ((uSingleSignificand & 0x400) == 0);
             uSingleSignificand &= HALF_SIGNIFICAND_MASK;
             uSingleSignificand <<= (SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
         } else {
             // Just zero
             uSingleBiasedExponent = SINGLE_EXPONENT_ZERO + SINGLE_EXPONENT_BIAS;
             uSingleSignificand = 0;
         }
     } else if(nHalfUnBiasedExponent == HALF_EXPONENT_INF_OR_NAN) {
         // NaN or Inifinity
         uSingleBiasedExponent = SINGLE_EXPONENT_INF_OR_NAN + SINGLE_EXPONENT_BIAS;
         if(uHalfSignificand) {
             // NaN
             // First preserve the NaN payload from half to single
             uSingleSignificand = uHalfSignificand & ~HALF_QUIET_NAN_BIT;
             if(uHalfSignificand & HALF_QUIET_NAN_BIT) {
                 // Next, set qNaN if needed since half qNaN bit is not copied above
                 uSingleSignificand |= SINGLE_QUIET_NAN_BIT;
             }
         } else {
             // Infinity
             uSingleSignificand = 0;
         }
     } else {
         // Normal number
         uSingleBiasedExponent = nHalfUnBiasedExponent + SINGLE_EXPONENT_BIAS;
         uSingleSignificand = uHalfSignificand << (SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
     }
     uSingleSign = uHalfSign;


     // Shift the three parts of the single precision into place
     const uint32_t uSinglePrecision = uSingleSignificand |
                                      (uSingleBiasedExponent << SINGLE_EXPONENT_SHIFT) |
                                      (uSingleSign << SINGLE_SIGN_SHIFT);

     return CopyUint32ToFloat(uSinglePrecision);
 }


 // Public function; see ieee754.h
 double IEEE754_HalfToDouble(uint16_t uHalfPrecision)
 {
     // Pull out the three parts of the half-precision float
     const uint16_t uHalfSignificand      =   uHalfPrecision & HALF_SIGNIFICAND_MASK;
     const int16_t  nHalfUnBiasedExponent = ((uHalfPrecision & HALF_EXPONENT_MASK) >> HALF_EXPONENT_SHIFT) - HALF_EXPONENT_BIAS;
     const uint16_t uHalfSign             =  (uHalfPrecision & HALF_SIGN_MASK) >> HALF_SIGN_SHIFT;


     // Make the three parts of hte single-precision number
     uint64_t uDoubleSignificand, uDoubleSign, uDoubleBiasedExponent;
     if(nHalfUnBiasedExponent == HALF_EXPONENT_ZERO) {
         // 0 or subnormal
         uDoubleBiasedExponent = DOUBLE_EXPONENT_ZERO + DOUBLE_EXPONENT_BIAS;
         if(uHalfSignificand) {
             // Subnormal case
             uDoubleBiasedExponent = -HALF_EXPONENT_BIAS + DOUBLE_EXPONENT_BIAS +1;
             // A half-precision subnormal can always be converted to a normal double-precision float because the ranges line up
             uDoubleSignificand = uHalfSignificand;
             // Shift bits from right of the decimal to left, reducing the exponent by 1 each time
             do {
                 uDoubleSignificand <<= 1;
                 uDoubleBiasedExponent--;
             } while ((uDoubleSignificand & 0x400) == 0);
             uDoubleSignificand &= HALF_SIGNIFICAND_MASK;
             uDoubleSignificand <<= (DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
         } else {
             // Just zero
             uDoubleSignificand = 0;
         }
     } else if(nHalfUnBiasedExponent == HALF_EXPONENT_INF_OR_NAN) {
         // NaN or Inifinity
         uDoubleBiasedExponent = DOUBLE_EXPONENT_INF_OR_NAN + DOUBLE_EXPONENT_BIAS;
         if(uHalfSignificand) {
             // NaN
             // First preserve the NaN payload from half to single
             uDoubleSignificand = uHalfSignificand & ~HALF_QUIET_NAN_BIT;
             if(uHalfSignificand & HALF_QUIET_NAN_BIT) {
                 // Next, set qNaN if needed since half qNaN bit is not copied above
                 uDoubleSignificand |= DOUBLE_QUIET_NAN_BIT;
             }
         } else {
             // Infinity
             uDoubleSignificand = 0;
         }
     } else {
         // Normal number
         uDoubleBiasedExponent = nHalfUnBiasedExponent + DOUBLE_EXPONENT_BIAS;
         uDoubleSignificand    = (uint64_t)uHalfSignificand << (DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
     }
     uDoubleSign = uHalfSign;


     // Shift the 3 parts into place as a double-precision
     const uint64_t uDouble = uDoubleSignificand |
                             (uDoubleBiasedExponent << DOUBLE_EXPONENT_SHIFT) |
                             (uDoubleSign << DOUBLE_SIGN_SHIFT);
     return CopyUint64ToDouble(uDouble);
 }


 // Public function; see ieee754.h
 IEEE754_union IEEE754_FloatToSmallest(float f)
 {
     IEEE754_union result;

     // Pull the neeed two parts out of the single-precision float
     const uint32_t uSingle = CopyFloatToUint32(f);
     const int32_t  nSingleExponent    = ((uSingle & SINGLE_EXPONENT_MASK) >> SINGLE_EXPONENT_SHIFT) - SINGLE_EXPONENT_BIAS;
     const uint32_t uSingleSignificand =   uSingle & SINGLE_SIGNIFICAND_MASK;

     // Bit mask that is the significand bits that would be lost when converting
     // from single-precision to half-precision
     const uint64_t uDroppedSingleBits = SINGLE_SIGNIFICAND_MASK >> HALF_NUM_SIGNIFICAND_BITS;

     // Optimizer will re organize so there is only one call to IEEE754_FloatToHalf()
     if(uSingle == 0) {
         // Value is 0.0000, not a a subnormal
         result.uSize = IEEE754_UNION_IS_HALF;
         result.uValue  = IEEE754_FloatToHalf(f);
     } else if(nSingleExponent == SINGLE_EXPONENT_INF_OR_NAN) {
         // NaN, +/- infinity
         result.uSize = IEEE754_UNION_IS_HALF;
         result.uValue  = IEEE754_FloatToHalf(f);
     } else if((nSingleExponent >= HALF_EXPONENT_MIN) && nSingleExponent <= HALF_EXPONENT_MAX && (!(uSingleSignificand & uDroppedSingleBits))) {
         // Normal number in exponent range and precision won't be lost
         result.uSize = IEEE754_UNION_IS_HALF;
         result.uValue  = IEEE754_FloatToHalf(f);
     } else {
         // Subnormal, exponent out of range, or precision will be lost
         result.uSize = IEEE754_UNION_IS_SINGLE;
         result.uValue  = uSingle;
     }

     return result;
 }

 // Public function; see ieee754.h
 IEEE754_union IEEE754_DoubleToSmallestInternal(double d, int bAllowHalfPrecision)
 {
     IEEE754_union result;

     // Pull the needed two parts out of the double-precision float
     const uint64_t uDouble = CopyDoubleToUint64(d);
     const int64_t  nDoubleExponent     = ((uDouble & DOUBLE_EXPONENT_MASK) >> DOUBLE_EXPONENT_SHIFT) - DOUBLE_EXPONENT_BIAS;
     const uint64_t uDoubleSignificand  =   uDouble & DOUBLE_SIGNIFICAND_MASK;

     // Masks to check whether dropped significand bits are zero or not
     const uint64_t uDroppedDoubleBits = DOUBLE_SIGNIFICAND_MASK >> HALF_NUM_SIGNIFICAND_BITS;
     const uint64_t uDroppedSingleBits = DOUBLE_SIGNIFICAND_MASK >> SINGLE_NUM_SIGNIFICAND_BITS;

     // The various cases
     if(d == 0.0) { // Take care of positive and negative zero
         // Value is 0.0000, not a a subnormal
         result.uSize  = IEEE754_UNION_IS_HALF;
         result.uValue = IEEE754_DoubleToHalf(d);
     } else if(nDoubleExponent == DOUBLE_EXPONENT_INF_OR_NAN) {
         // NaN, +/- infinity
         result.uSize  = IEEE754_UNION_IS_HALF;
         result.uValue = IEEE754_DoubleToHalf(d);
     } else if(bAllowHalfPrecision && (nDoubleExponent >= HALF_EXPONENT_MIN) && nDoubleExponent <= HALF_EXPONENT_MAX && (!(uDoubleSignificand & uDroppedDoubleBits))) {
         // Can convert to half without precision loss
         result.uSize  = IEEE754_UNION_IS_HALF;
         result.uValue = IEEE754_DoubleToHalf(d);
     } else if((nDoubleExponent >= SINGLE_EXPONENT_MIN) && nDoubleExponent <= SINGLE_EXPONENT_MAX && (!(uDoubleSignificand & uDroppedSingleBits))) {
         // Can convert to single without precision loss
         result.uSize  = IEEE754_UNION_IS_SINGLE;
         result.uValue = CopyFloatToUint32((float)d);
     } else {
         // Can't convert without precision loss
         result.uSize  = IEEE754_UNION_IS_DOUBLE;
         result.uValue = uDouble;
     }

     return result;
 }
	/*==============================================================================
	ieee754.c -- floating point conversion between half, double and single precision

	Copyright (c) 2018-2019, Laurence Lundblade. All rights reserved.

	SPDX-License-Identifier: BSD-3-Clause

	See BSD-3-Clause license in README.md

	Created on 7/23/18
	==============================================================================*/

	#include "ieee754.h"
	#include <string.h> // For memcpy()


	/*
	This code is written for clarity and verifiability, not for size, on the assumption
	that the optimizer will do a good job. The LLVM optimizer, -Os, does seem to do the
	job and the resulting object code is smaller from combining code for the many different
	cases (normal, subnormal, infinity, zero...) for the conversions.

	Dead stripping is also really helpful to get code size down when floating point
	encoding is not needed.

	This code works solely using shifts and masks and thus has no dependency on
	any math libraries. It can even work if the CPU doesn't have any floating
	point support, though that isn't the most useful thing to do.

	The memcpy() dependency is only for CopyFloatToUint32() and friends which only
	is needed to avoid type punning when converting the actual float bits to
	an unsigned value so the bit shifts and masks can work.
	*/

	/*
	The references used to write this code:

	- IEEE 754-2008, particularly section 3.6 and 6.2.1

	- https://en.wikipedia.org/wiki/IEEE_754 and subordinate pages

	- https://stackoverflow.com/questions/19800415/why-does-ieee-754-reserve-so-many-nan-values
	*/


	// ----- Half Precsion -----------
	#define HALF_NUM_SIGNIFICAND_BITS (10)
	#define HALF_NUM_EXPONENT_BITS (5)
	#define HALF_NUM_SIGN_BITS (1)

	#define HALF_SIGNIFICAND_SHIFT (0)
	#define HALF_EXPONENT_SHIFT (HALF_NUM_SIGNIFICAND_BITS)
	#define HALF_SIGN_SHIFT (HALF_NUM_SIGNIFICAND_BITS + HALF_NUM_EXPONENT_BITS)

	#define HALF_SIGNIFICAND_MASK (0x3ff) // The lower 10 bits // 0x03ff
	#define HALF_EXPONENT_MASK (0x1f << HALF_EXPONENT_SHIFT) // 0x7c00 5 bits of exponent
	#define HALF_SIGN_MASK (0x01 << HALF_SIGN_SHIFT) // // 0x80001 bit of sign
	#define HALF_QUIET_NAN_BIT (0x01 << (HALF_NUM_SIGNIFICAND_BITS-1)) // 0x0200

	/* Biased Biased Unbiased Use
	0x00 0 -15 0 and subnormal
	0x01 1 -14 Smallest normal exponent
	0x1e 30 15 Largest normal exponent
	0x1F 31 16 NaN and Infinity */
	#define HALF_EXPONENT_BIAS (15)
	#define HALF_EXPONENT_MAX (HALF_EXPONENT_BIAS) // 15 Unbiased
	#define HALF_EXPONENT_MIN (-HALF_EXPONENT_BIAS+1) // -14 Unbiased
	#define HALF_EXPONENT_ZERO (-HALF_EXPONENT_BIAS) // -15 Unbiased
	#define HALF_EXPONENT_INF_OR_NAN (HALF_EXPONENT_BIAS+1) // 16 Unbiased


	// ------ Single Precision --------
	#define SINGLE_NUM_SIGNIFICAND_BITS (23)
	#define SINGLE_NUM_EXPONENT_BITS (8)
	#define SINGLE_NUM_SIGN_BITS (1)

	#define SINGLE_SIGNIFICAND_SHIFT (0)
	#define SINGLE_EXPONENT_SHIFT (SINGLE_NUM_SIGNIFICAND_BITS)
	#define SINGLE_SIGN_SHIFT (SINGLE_NUM_SIGNIFICAND_BITS + SINGLE_NUM_EXPONENT_BITS)

	#define SINGLE_SIGNIFICAND_MASK (0x7fffffUL) // The lower 23 bits
	#define SINGLE_EXPONENT_MASK (0xffUL << SINGLE_EXPONENT_SHIFT) // 8 bits of exponent
	#define SINGLE_SIGN_MASK (0x01UL << SINGLE_SIGN_SHIFT) // 1 bit of sign
	#define SINGLE_QUIET_NAN_BIT (0x01UL << (SINGLE_NUM_SIGNIFICAND_BITS-1))

	/* Biased Biased Unbiased Use
	0x0000 0 -127 0 and subnormal
	0x0001 1 -126 Smallest normal exponent
	0x7f 127 0 1
	0xfe 254 127 Largest normal exponent
	0xff 255 128 NaN and Infinity */
	#define SINGLE_EXPONENT_BIAS (127)
	#define SINGLE_EXPONENT_MAX (SINGLE_EXPONENT_BIAS) // 127 unbiased
	#define SINGLE_EXPONENT_MIN (-SINGLE_EXPONENT_BIAS+1) // -126 unbiased
	#define SINGLE_EXPONENT_ZERO (-SINGLE_EXPONENT_BIAS) // -127 unbiased
	#define SINGLE_EXPONENT_INF_OR_NAN (SINGLE_EXPONENT_BIAS+1) // 128 unbiased


	// --------- Double Precision ----------
	#define DOUBLE_NUM_SIGNIFICAND_BITS (52)
	#define DOUBLE_NUM_EXPONENT_BITS (11)
	#define DOUBLE_NUM_SIGN_BITS (1)

	#define DOUBLE_SIGNIFICAND_SHIFT (0)
	#define DOUBLE_EXPONENT_SHIFT (DOUBLE_NUM_SIGNIFICAND_BITS)
	#define DOUBLE_SIGN_SHIFT (DOUBLE_NUM_SIGNIFICAND_BITS + DOUBLE_NUM_EXPONENT_BITS)

	#define DOUBLE_SIGNIFICAND_MASK (0xfffffffffffffULL) // The lower 52 bits
	#define DOUBLE_EXPONENT_MASK (0x7ffULL << DOUBLE_EXPONENT_SHIFT) // 11 bits of exponent
	#define DOUBLE_SIGN_MASK (0x01ULL << DOUBLE_SIGN_SHIFT) // 1 bit of sign
	#define DOUBLE_QUIET_NAN_BIT (0x01ULL << (DOUBLE_NUM_SIGNIFICAND_BITS-1))


	/* Biased Biased Unbiased Use
	0x00000000 0 -1023 0 and subnormal
	0x00000001 1 -1022 Smallest normal exponent
	0x000007fe 2046 1023 Largest normal exponent
	0x000007ff 2047 1024 NaN and Infinity */
	#define DOUBLE_EXPONENT_BIAS (1023)
	#define DOUBLE_EXPONENT_MAX (DOUBLE_EXPONENT_BIAS) // unbiased
	#define DOUBLE_EXPONENT_MIN (-DOUBLE_EXPONENT_BIAS+1) // unbiased
	#define DOUBLE_EXPONENT_ZERO (-DOUBLE_EXPONENT_BIAS) // unbiased
	#define DOUBLE_EXPONENT_INF_OR_NAN (DOUBLE_EXPONENT_BIAS+1) // unbiased



	/*
	Convenient functions to avoid type punning, compiler warnings and such
	The optimizer reduces them to a simple assignment.
	This is a crusty corner of C. It shouldn't be this hard.

	These are also in UsefulBuf.h under a different name. They are copied
	here to avoid a dependency on UsefulBuf.h. There is no
	object code size impact because these always optimze down to a
	simple assignment.
	*/
	static inline uint32_t CopyFloatToUint32(float f)
	{
	uint32_t u32;
	memcpy(&u32, &f, sizeof(uint32_t));
	return u32;
	}

	static inline uint64_t CopyDoubleToUint64(double d)
	{
	uint64_t u64;
	memcpy(&u64, &d, sizeof(uint64_t));
	return u64;
	}

	static inline float CopyUint32ToFloat(uint32_t u32)
	{
	float f;
	memcpy(&f, &u32, sizeof(uint32_t));
	return f;
	}

	static inline double CopyUint64ToDouble(uint64_t u64)
	{
	double d;
	memcpy(&d, &u64, sizeof(uint64_t));
	return d;
	}


	// Public function; see ieee754.h
	uint16_t IEEE754_FloatToHalf(float f)
	{
	// Pull the three parts out of the single-precision float
	const uint32_t uSingle = CopyFloatToUint32(f);
	const int32_t nSingleUnbiasedExponent = ((uSingle & SINGLE_EXPONENT_MASK) >> SINGLE_EXPONENT_SHIFT) - SINGLE_EXPONENT_BIAS;
	const uint32_t uSingleSign = (uSingle & SINGLE_SIGN_MASK) >> SINGLE_SIGN_SHIFT;
	const uint32_t uSingleSignificand = uSingle & SINGLE_SIGNIFICAND_MASK;


	// Now convert the three parts to half-precision.
	uint16_t uHalfSign, uHalfSignificand, uHalfBiasedExponent;
	if(nSingleUnbiasedExponent == SINGLE_EXPONENT_INF_OR_NAN) {
	// +/- Infinity and NaNs -- single biased exponent is 0xff
	uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
	if(!uSingleSignificand) {
	// Infinity
	uHalfSignificand = 0;
	} else {
	// Copy the LBSs of the NaN payload that will fit from the single to the half
	uHalfSignificand = uSingleSignificand & (HALF_SIGNIFICAND_MASK & ~HALF_QUIET_NAN_BIT);
	if(uSingleSignificand & SINGLE_QUIET_NAN_BIT) {
	// It's a qNaN; copy the qNaN bit
	uHalfSignificand \|= HALF_QUIET_NAN_BIT;
	} else {
	// It's a sNaN; make sure the significand is not zero so it stays a NaN
	// This is needed because not all significand bits are copied from single
	if(!uHalfSignificand) {
	// Set the LSB. This is what wikipedia shows for sNAN.
	uHalfSignificand \|= 0x01;
	}
	}
	}
	} else if(nSingleUnbiasedExponent == SINGLE_EXPONENT_ZERO) {
	// 0 or a subnormal number -- singled biased exponent is 0
	uHalfBiasedExponent = 0;
	uHalfSignificand = 0; // Any subnormal single will be too small to express as a half precision
	} else if(nSingleUnbiasedExponent > HALF_EXPONENT_MAX) {
	// Exponent is too large to express in half-precision; round up to infinity
	uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
	uHalfSignificand = 0;
	} else if(nSingleUnbiasedExponent < HALF_EXPONENT_MIN) {
	// Exponent is too small to express in half-precision normal; make it a half-precision subnormal
	uHalfBiasedExponent = (uint16_t)(HALF_EXPONENT_ZERO + HALF_EXPONENT_BIAS);
	// Difference between single normal exponent and the base exponent of a half subnormal
	const uint32_t nExpDiff = -(nSingleUnbiasedExponent - HALF_EXPONENT_MIN);
	// Also have to shift the significand by the difference in number of bits between a single and a half significand
	const int32_t nSignificandBitsDiff = SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS;
	// Add in the 1 that is implied in the significand of a normal number; it needs to be present in a subnormal
	const uint32_t uSingleSignificandSubnormal = uSingleSignificand + (0x01L << SINGLE_NUM_SIGNIFICAND_BITS);
	uHalfSignificand = uSingleSignificandSubnormal >> (nExpDiff + nSignificandBitsDiff);
	} else {
	// The normal case
	uHalfBiasedExponent = nSingleUnbiasedExponent + HALF_EXPONENT_BIAS;
	uHalfSignificand = uSingleSignificand >> (SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
	}
	uHalfSign = uSingleSign;

	// Put the 3 values in the right place for a half precision
	const uint16_t uHalfPrecision = uHalfSignificand \|
	(uHalfBiasedExponent << HALF_EXPONENT_SHIFT) \|
	(uHalfSign << HALF_SIGN_SHIFT);
	return uHalfPrecision;
	}


	// Public function; see ieee754.h
	uint16_t IEEE754_DoubleToHalf(double d)
	{
	// Pull the three parts out of the double-precision float
	const uint64_t uDouble = CopyDoubleToUint64(d);
	const int64_t nDoubleUnbiasedExponent = ((uDouble & DOUBLE_EXPONENT_MASK) >> DOUBLE_EXPONENT_SHIFT) - DOUBLE_EXPONENT_BIAS;
	const uint64_t uDoubleSign = (uDouble & DOUBLE_SIGN_MASK) >> DOUBLE_SIGN_SHIFT;
	const uint64_t uDoubleSignificand = uDouble & DOUBLE_SIGNIFICAND_MASK;


	// Now convert the three parts to half-precision.
	uint16_t uHalfSign, uHalfSignificand, uHalfBiasedExponent;
	if(nDoubleUnbiasedExponent == DOUBLE_EXPONENT_INF_OR_NAN) {
	// +/- Infinity and NaNs -- single biased exponent is 0xff
	uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
	if(!uDoubleSignificand) {
	// Infinity
	uHalfSignificand = 0;
	} else {
	// Copy the LBSs of the NaN payload that will fit from the double to the half
	uHalfSignificand = uDoubleSignificand & (HALF_SIGNIFICAND_MASK & ~HALF_QUIET_NAN_BIT);
	if(uDoubleSignificand & DOUBLE_QUIET_NAN_BIT) {
	// It's a qNaN; copy the qNaN bit
	uHalfSignificand \|= HALF_QUIET_NAN_BIT;
	} else {
	// It's an sNaN; make sure the significand is not zero so it stays a NaN
	// This is needed because not all significand bits are copied from single
	if(!uHalfSignificand) {
	// Set the LSB. This is what wikipedia shows for sNAN.
	uHalfSignificand \|= 0x01;
	}
	}
	}
	} else if(nDoubleUnbiasedExponent == DOUBLE_EXPONENT_ZERO) {
	// 0 or a subnormal number -- double biased exponent is 0
	uHalfBiasedExponent = 0;
	uHalfSignificand = 0; // Any subnormal single will be too small to express as a half precision; TODO, is this really true?
	} else if(nDoubleUnbiasedExponent > HALF_EXPONENT_MAX) {
	// Exponent is too large to express in half-precision; round up to infinity; TODO, is this really true?
	uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
	uHalfSignificand = 0;
	} else if(nDoubleUnbiasedExponent < HALF_EXPONENT_MIN) {
	// Exponent is too small to express in half-precision; round down to zero
	uHalfBiasedExponent = (uint16_t)(HALF_EXPONENT_ZERO + HALF_EXPONENT_BIAS);
	// Difference between double normal exponent and the base exponent of a half subnormal
	const uint64_t nExpDiff = -(nDoubleUnbiasedExponent - HALF_EXPONENT_MIN);
	// Also have to shift the significand by the difference in number of bits between a double and a half significand
	const int64_t nSignificandBitsDiff = DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS;
	// Add in the 1 that is implied in the significand of a normal number; it needs to be present in a subnormal
	const uint64_t uDoubleSignificandSubnormal = uDoubleSignificand + (0x01ULL << DOUBLE_NUM_SIGNIFICAND_BITS);
	uHalfSignificand = uDoubleSignificandSubnormal >> (nExpDiff + nSignificandBitsDiff);
	} else {
	// The normal case
	uHalfBiasedExponent = nDoubleUnbiasedExponent + HALF_EXPONENT_BIAS;
	uHalfSignificand = uDoubleSignificand >> (DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
	}
	uHalfSign = uDoubleSign;


	// Put the 3 values in the right place for a half precision
	const uint16_t uHalfPrecision = uHalfSignificand \|
	(uHalfBiasedExponent << HALF_EXPONENT_SHIFT) \|
	(uHalfSign << HALF_SIGN_SHIFT);
	return uHalfPrecision;
	}


	// Public function; see ieee754.h
	float IEEE754_HalfToFloat(uint16_t uHalfPrecision)
	{
	// Pull out the three parts of the half-precision float
	const uint16_t uHalfSignificand = uHalfPrecision & HALF_SIGNIFICAND_MASK;
	const int16_t nHalfUnBiasedExponent = ((uHalfPrecision & HALF_EXPONENT_MASK) >> HALF_EXPONENT_SHIFT) - HALF_EXPONENT_BIAS;
	const uint16_t uHalfSign = (uHalfPrecision & HALF_SIGN_MASK) >> HALF_SIGN_SHIFT;


	// Make the three parts of the single-precision number
	uint32_t uSingleSignificand, uSingleSign, uSingleBiasedExponent;
	if(nHalfUnBiasedExponent == HALF_EXPONENT_ZERO) {
	// 0 or subnormal
	if(uHalfSignificand) {
	// Subnormal case
	uSingleBiasedExponent = -HALF_EXPONENT_BIAS + SINGLE_EXPONENT_BIAS +1;
	// A half-precision subnormal can always be converted to a normal single-precision float because the ranges line up
	uSingleSignificand = uHalfSignificand;
	// Shift bits from right of the decimal to left, reducing the exponent by 1 each time
	do {
	uSingleSignificand <<= 1;
	uSingleBiasedExponent--;
	} while ((uSingleSignificand & 0x400) == 0);
	uSingleSignificand &= HALF_SIGNIFICAND_MASK;
	uSingleSignificand <<= (SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
	} else {
	// Just zero
	uSingleBiasedExponent = SINGLE_EXPONENT_ZERO + SINGLE_EXPONENT_BIAS;
	uSingleSignificand = 0;
	}
	} else if(nHalfUnBiasedExponent == HALF_EXPONENT_INF_OR_NAN) {
	// NaN or Inifinity
	uSingleBiasedExponent = SINGLE_EXPONENT_INF_OR_NAN + SINGLE_EXPONENT_BIAS;
	if(uHalfSignificand) {
	// NaN
	// First preserve the NaN payload from half to single
	uSingleSignificand = uHalfSignificand & ~HALF_QUIET_NAN_BIT;
	if(uHalfSignificand & HALF_QUIET_NAN_BIT) {
	// Next, set qNaN if needed since half qNaN bit is not copied above
	uSingleSignificand \|= SINGLE_QUIET_NAN_BIT;
	}
	} else {
	// Infinity
	uSingleSignificand = 0;
	}
	} else {
	// Normal number
	uSingleBiasedExponent = nHalfUnBiasedExponent + SINGLE_EXPONENT_BIAS;
	uSingleSignificand = uHalfSignificand << (SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
	}
	uSingleSign = uHalfSign;


	// Shift the three parts of the single precision into place
	const uint32_t uSinglePrecision = uSingleSignificand \|
	(uSingleBiasedExponent << SINGLE_EXPONENT_SHIFT) \|
	(uSingleSign << SINGLE_SIGN_SHIFT);

	return CopyUint32ToFloat(uSinglePrecision);
	}


	// Public function; see ieee754.h
	double IEEE754_HalfToDouble(uint16_t uHalfPrecision)
	{
	// Pull out the three parts of the half-precision float
	const uint16_t uHalfSignificand = uHalfPrecision & HALF_SIGNIFICAND_MASK;
	const int16_t nHalfUnBiasedExponent = ((uHalfPrecision & HALF_EXPONENT_MASK) >> HALF_EXPONENT_SHIFT) - HALF_EXPONENT_BIAS;
	const uint16_t uHalfSign = (uHalfPrecision & HALF_SIGN_MASK) >> HALF_SIGN_SHIFT;


	// Make the three parts of hte single-precision number
	uint64_t uDoubleSignificand, uDoubleSign, uDoubleBiasedExponent;
	if(nHalfUnBiasedExponent == HALF_EXPONENT_ZERO) {
	// 0 or subnormal
	uDoubleBiasedExponent = DOUBLE_EXPONENT_ZERO + DOUBLE_EXPONENT_BIAS;
	if(uHalfSignificand) {
	// Subnormal case
	uDoubleBiasedExponent = -HALF_EXPONENT_BIAS + DOUBLE_EXPONENT_BIAS +1;
	// A half-precision subnormal can always be converted to a normal double-precision float because the ranges line up
	uDoubleSignificand = uHalfSignificand;
	// Shift bits from right of the decimal to left, reducing the exponent by 1 each time
	do {
	uDoubleSignificand <<= 1;
	uDoubleBiasedExponent--;
	} while ((uDoubleSignificand & 0x400) == 0);
	uDoubleSignificand &= HALF_SIGNIFICAND_MASK;
	uDoubleSignificand <<= (DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
	} else {
	// Just zero
	uDoubleSignificand = 0;
	}
	} else if(nHalfUnBiasedExponent == HALF_EXPONENT_INF_OR_NAN) {
	// NaN or Inifinity
	uDoubleBiasedExponent = DOUBLE_EXPONENT_INF_OR_NAN + DOUBLE_EXPONENT_BIAS;
	if(uHalfSignificand) {
	// NaN
	// First preserve the NaN payload from half to single
	uDoubleSignificand = uHalfSignificand & ~HALF_QUIET_NAN_BIT;
	if(uHalfSignificand & HALF_QUIET_NAN_BIT) {
	// Next, set qNaN if needed since half qNaN bit is not copied above
	uDoubleSignificand \|= DOUBLE_QUIET_NAN_BIT;
	}
	} else {
	// Infinity
	uDoubleSignificand = 0;
	}
	} else {
	// Normal number
	uDoubleBiasedExponent = nHalfUnBiasedExponent + DOUBLE_EXPONENT_BIAS;
	uDoubleSignificand = (uint64_t)uHalfSignificand << (DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
	}
	uDoubleSign = uHalfSign;


	// Shift the 3 parts into place as a double-precision
	const uint64_t uDouble = uDoubleSignificand \|
	(uDoubleBiasedExponent << DOUBLE_EXPONENT_SHIFT) \|
	(uDoubleSign << DOUBLE_SIGN_SHIFT);
	return CopyUint64ToDouble(uDouble);
	}


	// Public function; see ieee754.h
	IEEE754_union IEEE754_FloatToSmallest(float f)
	{
	IEEE754_union result;

	// Pull the neeed two parts out of the single-precision float
	const uint32_t uSingle = CopyFloatToUint32(f);
	const int32_t nSingleExponent = ((uSingle & SINGLE_EXPONENT_MASK) >> SINGLE_EXPONENT_SHIFT) - SINGLE_EXPONENT_BIAS;
	const uint32_t uSingleSignificand = uSingle & SINGLE_SIGNIFICAND_MASK;

	// Bit mask that is the significand bits that would be lost when converting
	// from single-precision to half-precision
	const uint64_t uDroppedSingleBits = SINGLE_SIGNIFICAND_MASK >> HALF_NUM_SIGNIFICAND_BITS;

	// Optimizer will re organize so there is only one call to IEEE754_FloatToHalf()
	if(uSingle == 0) {
	// Value is 0.0000, not a a subnormal
	result.uSize = IEEE754_UNION_IS_HALF;
	result.uValue = IEEE754_FloatToHalf(f);
	} else if(nSingleExponent == SINGLE_EXPONENT_INF_OR_NAN) {
	// NaN, +/- infinity
	result.uSize = IEEE754_UNION_IS_HALF;
	result.uValue = IEEE754_FloatToHalf(f);
	} else if((nSingleExponent >= HALF_EXPONENT_MIN) && nSingleExponent <= HALF_EXPONENT_MAX && (!(uSingleSignificand & uDroppedSingleBits))) {
	// Normal number in exponent range and precision won't be lost
	result.uSize = IEEE754_UNION_IS_HALF;
	result.uValue = IEEE754_FloatToHalf(f);
	} else {
	// Subnormal, exponent out of range, or precision will be lost
	result.uSize = IEEE754_UNION_IS_SINGLE;
	result.uValue = uSingle;
	}

	return result;
	}

	// Public function; see ieee754.h
	IEEE754_union IEEE754_DoubleToSmallestInternal(double d, int bAllowHalfPrecision)
	{
	IEEE754_union result;

	// Pull the needed two parts out of the double-precision float
	const uint64_t uDouble = CopyDoubleToUint64(d);
	const int64_t nDoubleExponent = ((uDouble & DOUBLE_EXPONENT_MASK) >> DOUBLE_EXPONENT_SHIFT) - DOUBLE_EXPONENT_BIAS;
	const uint64_t uDoubleSignificand = uDouble & DOUBLE_SIGNIFICAND_MASK;

	// Masks to check whether dropped significand bits are zero or not
	const uint64_t uDroppedDoubleBits = DOUBLE_SIGNIFICAND_MASK >> HALF_NUM_SIGNIFICAND_BITS;
	const uint64_t uDroppedSingleBits = DOUBLE_SIGNIFICAND_MASK >> SINGLE_NUM_SIGNIFICAND_BITS;

	// The various cases
	if(d == 0.0) { // Take care of positive and negative zero
	// Value is 0.0000, not a a subnormal
	result.uSize = IEEE754_UNION_IS_HALF;
	result.uValue = IEEE754_DoubleToHalf(d);
	} else if(nDoubleExponent == DOUBLE_EXPONENT_INF_OR_NAN) {
	// NaN, +/- infinity
	result.uSize = IEEE754_UNION_IS_HALF;
	result.uValue = IEEE754_DoubleToHalf(d);
	} else if(bAllowHalfPrecision && (nDoubleExponent >= HALF_EXPONENT_MIN) && nDoubleExponent <= HALF_EXPONENT_MAX && (!(uDoubleSignificand & uDroppedDoubleBits))) {
	// Can convert to half without precision loss
	result.uSize = IEEE754_UNION_IS_HALF;
	result.uValue = IEEE754_DoubleToHalf(d);
	} else if((nDoubleExponent >= SINGLE_EXPONENT_MIN) && nDoubleExponent <= SINGLE_EXPONENT_MAX && (!(uDoubleSignificand & uDroppedSingleBits))) {
	// Can convert to single without precision loss
	result.uSize = IEEE754_UNION_IS_SINGLE;
	result.uValue = CopyFloatToUint32((float)d);
	} else {
	// Can't convert without precision loss
	result.uSize = IEEE754_UNION_IS_DOUBLE;
	result.uValue = uDouble;
	}

	return result;
	}